Problem 21 - College Physics

Question

An electron in an atom is in an orbit that has a magnetic quantum number of \(m_{\ell}=2\). What are the minimum values that (a) \(\ell\) and (b) \(n\) could be for that orbit?

Step-by-Step Solution

Answer

Minimum \(\ell\) is 2; minimum \(n\) is 3.

Step 1: Understanding Quantum Numbers

Quantum numbers are used to describe the properties of atomic orbitals and the electrons in those orbitals. Here, the magnetic quantum number \(m_{\ell}\) is given as \(2\). We need to find the minimum possible values for the angular momentum quantum number \(\ell\) and the principal quantum number \(n\).

Step 2: Determining Minimum \(\ell\)

The magnetic quantum number \(m_{\ell}\) can take any integer value from \(-\ell\) to \(\ell\). Given that \(m_{\ell} = 2\), this means \(\ell\) must be at least \(2\) because \(m_{\ell}\) cannot exceed \(\ell\). Therefore, the minimum value of \(\ell\) is \(2\).

Step 3: Determining Minimum \(n\)

The principal quantum number \(n\) must be greater than \(\ell\) because \(n\) corresponds to the energy level or shell of the atom, and \(\ell\) represents the shape of the orbital within that shell (where \(\ell = 0, 1, ... , n-1\)). Since \(\ell = 2\) is the minimum, this implies \(n\) must be at least \(3\). Therefore, the minimum value of \(n\) is \(3\).

Key Concepts

Magnetic Quantum Number

The magnetic quantum number, symbolized as \( m_{\ell} \), is a crucial component of quantum mechanics.
It describes the orientation of an electron's orbital within a magnetic field. The values \( m_{\ell} \) can take on are integers, varying from \(-\ell\) to \(\ell\), where \(\ell\) is the angular momentum quantum number.
This means that \( m_{\ell} \) is directly linked to the shape and orientation of the orbital an electron occupies.

• Range of Values: \( m_{\ell} \) can range from \(-\ell\) to \(\ell\). For instance, if \(\ell = 2\), then \( m_{\ell} \) could be -2, -1, 0, 1, or 2.
• Physical Interpretation: The different \( m_{\ell} \) values correspond to the electron's different angular positions in space.

Understanding \( m_{\ell} \) is important for visually appreciating how electrons orient themselves in the atom due to external magnetic influences, giving rise to spectral line splitting noted in Zeeman effect studies.

Angular Momentum Quantum Number

The angular momentum quantum number, represented as \( \ell \), specifies the shape of the electron's orbital.
Think of it as the narrative of how the orbital where an electron resides appears—important for understanding atomic electron configurations.

• Range of \( \ell \): The \( \ell \) values range from 0 to \( n-1 \), where \( n \) is the principal quantum number.
• Orbital Types: Different \( \ell \) values describe distinctly shaped orbitals, such as s, p, d, and f orbitals, corresponding to \( \ell = 0, 1, 2, \) and 3, respectively.
• Minimum \( \ell \) in the Problem: Given \( m_{\ell} = 2\), the minimum \( \ell \) must be 2, as \( m_{\ell} \) cannot exceed \( \ell \).

By understanding \( \ell \), we can grasp which types of orbitals electrons can occupy and how these influence the chemical properties.

Principal Quantum Number

The principal quantum number, denoted by \( n \), primarily determines the energy level and size of an electron's orbital.
It can only take on positive integer values (1, 2, 3, ...), highlighting different energy levels or shells within an atom.

• Energy Levels: Higher values of \( n \) translate to larger orbitals that are further away from the nucleus.
• Relationship with \( \ell \): The \( n \) value must always be greater than the angular momentum quantum number \( \ell \) (since \( \ell \) ranges from 0 to \( n-1 \)).
• Minimum \( n \) in the Problem: With the minimum \( \ell \) determined as 2, \( n \) must be at least 3, ensuring a valid range for \( \ell \).

Recognizing the significance of \( n \) helps in defining the overall structure of the atom and predicting possible electron configurations essential for chemical interactions.